EXPONENTS. EXPONENTIAL NOTATION X IS THE BASE 2 IS THE EXPONENT OR POWER.

EXPONENTS

EXPONENTIAL NOTATION

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X 2

X IS THE BASE

2 IS THE EXPONENTOR POWER

EXPONENTIAL NOTATION

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X 2 = X • XTHE BASE IS SQUARED

EXPONENTIAL NOTATION

EXPONENT IS THE NUMBER OF TIMES THE BASE IS MULTIPLIED BY ITSELF

EXPONENTIAL NOTATION

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Y 8

Y IS THE BASE

8 IS THE EXPONENT

EXPONENTIAL NOTATION

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Y 8 = Y • Y • Y • Y • Y • Y • Y • Y

Y IS THE BASE

8 IS THE EXPONENT

Compare these two cases

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(−6)2 = −6• −6 = +36

−62 = −(6)2 = −(6) • (6) = −36

6 IS THE BASE(THE NEG. SIGN IS NOT PARTOF THE BASE, IT MUST REMAIN PART OF THE ANSWER)

-6 IS THE BASE

EXPONENTIAL NOTATION

€

(−h )2

WHAT IS THE BASE?

WHAT WILL BE SQUARED?

EXPONENTIAL NOTATION

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(−h )2

WHAT IS THE BASE?

-h is the base

EXPONENTIAL NOTATION

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−6r5

WHAT IS THE BASE?

What will be raised to the 5th power?

EXPONENTIAL NOTATION

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−6r5

WHAT IS THE BASE?

r will be raised to the 5th power

TRY THESE

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(−5)3 =

−53 =

TRY THESE

€

(−5)3 = −5• −5• −5 = −125

−53 = −5• 5 • 5 = −125

EVALUATING EXPRESSIONS WITH EXPONENTS

Evaluating Expression with Exponents

Evaluate 2x2(x+y)When x=6 & y=3

Evaluating Expression with Exponents

Evaluate 2x2(x+y)When x=6 & y=3

1. Put in x & y values

2x2(x+y)= 2(6)2(6+3)

Evaluating Expression with Exponents

Evaluate 2x2(x+y)When x=6 & y=3

1. Put in x & y values

2. Use PEMDAS2x2(x+y)= 2(6)2(6+3)= 2(6)2(9)= 2(36)9= 72•9= 648

MULTIPLYING SIMILAR BASES

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X1 • X1 = X1+1 = X 2

XX2

X

MULTIPLYING SIMILAR BASES

THE RULE IS TO ADD THE EXPONENTS

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X 2 • X 3 = X 2+3 = X 5

X a • X b = X a +b

MULTIPLYING SIMILAR BASES

€

X1 • X1 = ?

€

X 2 • X 3 = ?

X a • X b = ?

MULTIPLYING SIMILAR BASES

€

X1 • X1 = X1+1 = X 2

€

X 2 • X 3 = X 2+3 = X 5

X a • X b = X a +b

TRY THESE PROBLEMS

€

X 2 • X 7 =

€

X−2 • X 3 =

X−5 • X−4 =

DIVIDING SIMILAR BASES

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X 2

X=

X • X

X= X

DIVIDING SIMILAR BASES

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X 5

X 2=

X • X • X • X • X

X • X= X • X • X = X 3

THE RULE IS TO SUBTRACT THE EXPONENT OF

THE DENOMINATOR FROM THE EXPONENT OF

THE NUMERATOR

DIVIDING SIMILAR BASES

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X a

X b= X a−b

X 6

X 2= X 6−2 = X 4

X 7

X−5= X 7−(−5) = X12

THE RULE IS TO SUBTRACT THE EXPONENTS

TRY THESE

€

X 8

X 6=

X 5

X−9=

Y −7

Y −10=

TRY THESE

€

X 8

X 6= X 8−6 = X 2

X 5

X−9= X 5−(−9) = X14

Y −7

Y −10= Y −7−(−10) = Y 3

NEGATIVE EXPONENTS

€

Y −4 =1

Y 4

1

Y −4= Y 4

€

X−2 =1

X 2

1

X−2= X 2

You can change a negative exponent to positive by switching it’s base from numerator to denominator or vice versa.

NEGATIVE EXPONENTS

€

X−2 =1

X 2

MOVE THE BASE & EXPONENT FROM THE NUMERATOR TO THE

DENOMINATOR OR VICE VERSA AND CHANGE THE SIGN OF THE EXPONENT

NEGATIVE EXPONENTS

X2/X2 IS WHAT PROPERTY?

€

X−2 = X−2 X 2

X 2=

€

X−2 X 2

X 2

⎛

⎝ ⎜

⎞

⎠ ⎟=

X−2 • X 2

1• X 2

MULTIPLY NUMERATORS & MULTIPLY DENOMINATORS

NEGATIVE EXPONENTS

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X−2 X 2

X 2

⎛

⎝ ⎜

⎞

⎠ ⎟=

X−2 • X 2

1• X 2=

X−2+2

X 2=

X 0

X 2

ANYTHING TO THE ZERO POWER IS EQUAL TO ?

NEGATIVE EXPONENTS

Why is anything to the zero power equal to 1?

Check Out These Patterns

€

23 = 8

22 = 4 = (8 ÷ 2)

21 = 2 = (4 ÷ 2)

20 =1 = (2 ÷ 2)

2−1 =1

2= (1÷ 2)

€

53 =125

52 = 25 = (125 ÷ 5)

51 = 5 = (25 ÷ 5)

50 =1= (5 ÷ 5)

5−1 =1

5= (1÷ 5)

Or For Anything

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x 3 = x • x • x = x 4

x( )

x 2 = x • x = x 3

x( )

x1 = x = x 2

x( )

x 0 =1= xx( )

x−1 =1

x= 1

x( )

€

X−2 X 2

X 2

⎛

⎝ ⎜

⎞

⎠ ⎟=

X−2 • X 2

1• X 2=

X−2+2

X 2=

X 0

X 2=

1

X 2

THIS IS WHY

SWITCHING A NEGATIVE EXPONENT CHANGES ITS SIGN

Let’s compare the Neg. ExponentRule with the Dividing Fraction Rule

To divide fractions you

INVERT THE 2ND FRACTION AND CHANGE THE DIVISION SIGN TO MULTIPICATION

For negative exponents you

INVERT THE BASE WITH AND CHANGE THE SIGN OF THE EXPONENT

Try These

€

x−4 =

y−8 =

2x−9 =

Try These

€

x−4 =1

x 4

y−8 =1

y 8

2x−9 =2

x 9

What if the negative exponent is in the denominator?

€

1

x−2=

The same rule of inverting the base with the exponent and making the exponent positive applies but let see why this is so.

What if the negative exponent is in the denominator?

€

1

x−2=

1

x−2

x 2

x 2

⎛

⎝ ⎜

⎞

⎠ ⎟=

x 2

x−2+2=

Use the Multiplicative Identity to Simplify

What if the negative exponent is in the denominator?

€

1

x−2=

1

x−2

x 2

x 2

⎛

⎝ ⎜

⎞

⎠ ⎟=

x 2

x−2+2=

x 2

x 0=

Do you remember what x0 is equal to?

What if the negative exponent is in the denominator?

€

1

x−2=

1

x−2

x 2

x 2

⎛

⎝ ⎜

⎞

⎠ ⎟=

x 2

x−2+2=

x 2

x 0=

x 2

1= x 2

INVERT AND CHANGE EXPONENT SIGN

Try These

€

1

y−5=

2

3x−10=

x−3

y−8=

Try These ANSWERS

€

1

y−5= y 5

2

3x−10=

2x10

3

x−3

y−8=

y 8

x 3

SUMMARY SO FAREXPONENTIAL NOTATION

€

X

2

X IS THE BASE

2 IS THE EXPONENT

€

X−2 =1

X 2

1

X−2= X 2

DIVIDING EXPONENTS

SUBTRACT THE EXPONENT OF THE DENOMINATOR FROM THE

EXPONENT OF THE NUMERATOR

MULTIPLYING EXPONENTS

ADD THE EXPONENTS

€

X A • X B = X A +B

€

X A

X B= X A−B

INVERTING A NEGATIVE EXPONENT CHANGES ITS SIGN

DON’T BE MARY TO THE Z POWER

GET WITH IT!

POWER TO A POWER

€

X 2( )

3= X 2

( ) X 2( ) X 2

( ) = X 2+2+2 = X 6

THIS IS A POWERFUL IDEA

POWER TO A POWER VS. MULT. SIMILAR BASES

€

X 2 • X 3 = X 2+3 = X 5

X 2( )

3= X 2

( ) X 2( ) X 2

( ) = X 2+2+2 = X 6

POWER TO A POWER

€

X 2( )

3= X 2

( ) X 2( ) X 2

( )

= X • X( ) X • X( ) X • X( ) = X 6

MULTIPLY THE EXPONENTS

€

X a( )

b= X a⋅b

POWER TO A POWER

€

53( )

4=

D9( )

3=

a−5( )

6=

Express Answers as Positive Exponents

POWER TO A POWER

€

53( )

4= 53• 4 = 512

D9( )

3= D9• 3 = D27

a−5( )

6= a−5• 6 =a−30=

1

a30

COMPARINGMULTIPLY SIMILAR BASES

& POWER TO A POWER

€

X A • X B = X A +B

€

X A( )

B= X A• B

WHAT’S THE DIFFERENCE?

€

X A • Y B( )

C=

€

AB( )C

=

WHAT’S THE DIFFERENCE?

€

X A • Y B( )

C= XC • AY C • B

€

AB( )C

= AC BC

NOTHING REALLY(A MONOMIAL IS A PRODUCT)

A monomial is a term with a number and one or more variables (letters) raised to some power. Examples are:

€

5x

a

−23a5

2,345m2n−5 p23

€

X A • Y B( )

C= XC • AY C • B

€

AB( )C

= AC BC

€

AB( )C

= AC BC

€

AB( )C

= A1B1( )

C= A1•C B1•C = AC BC

Let’s take a close look at DISTRIBUTING THE POWER

Simplify (4d5)2

Powerof a Monomial

Simplify (4d5)2

= 41•2 • d5•2

= 42 d10

1. Multiply the outsidepower to the insidepowers.

Powerof a Monomial

Simplify (4d5)2

= 41•2 • d5•2

= 42 d10

= 16 d10

1. Multiply the outsidepower to the insidepowers.

2. Simplify

Powerof a Monomial

Power of a MonomialSimplify (2x3y4)5

1. Multiply the outsidepower to the insidepowers.

2. Simplify

Power of a MonomialSimplify (2x3y4)5

= 21•5 • x3•5•y4•5

= 25 x15y20

or 32 x15y20

1. Multiply the outsidepower to the insidepowers.

2. Simplify

Take a Powerof a QuotientSimplify

€

3x 3y

7z2

⎛

⎝ ⎜

⎞

⎠ ⎟

4

Take a Powerof a QuotientSimplify

1. Multiply the outsidepower to the insidepowers.

€

3x 3y

7z2

⎛

⎝ ⎜

⎞

⎠ ⎟

4

€

3x 3y

2z2

⎛

⎝ ⎜

⎞

⎠ ⎟

4

=34 x 3•4 y1•4

24 z2•4

=34 x12y 4

24 z8=

Take a Powerof a QuotientSimplify

1. Multiply the outsidepower to the insidepowers.

2. Simplify

€

3x 3y

7z2

⎛

⎝ ⎜

⎞

⎠ ⎟

4

€

3x 3y

2z2

⎛

⎝ ⎜

⎞

⎠ ⎟

4

=34 x 3•4 y1•4

24 z2•4

=34 x12y 4

24 z8=

81x12y 4

16z8

Mult/Divide Monomials with

Exponents

€

3x 3y 5( ) • −8x 2y( )

Mult/Divide Monomials with

Exponents

1. Multiply the numbers2. Multiply the Similar

Variables by adding the exponents.

€

3x 3y 5( ) • −8x 2y( )

€

3x 3y 5( ) • −8x 2y( )

= 3 • −8x 3+2y 5+1

Mult/Divide Monomials with

Exponents

1. Multiply the numbers2. Multiply the Similar

Variables by adding the exponents.

3. Simplify

€

3x 3y 5( ) • −8x 2y( )

€

3x 3y 5( ) • −8x 2y( )

= 3 • −8x 3+2y 5+1

= −24 x 5y 6

5 Exponent Rules1.When multiplying similar bases add exponents

2.When dividing similar bases subtract exponents

3.When inverting change exponents’ sign

4.When taking a power of a power multiply exponents

5.When taking a power of a product or quotient distribute the outside power to the inside powers

EXPONENTIAL NOTATION

€

X

2

X IS THE BASE

2 IS THE EXPONENT

6 IS THE BASE( . THE NEG SIGN IS NOT PART

, OF THE BASE IT MUST REMAIN )PART OF THE ANSWER

-6 IS THE BASE

Compare these two cases

€

( − 6 )

2

= − 6 • − 6 = + 36

− 6

2

= − ( 6 )

2

= − ( 6 ) • ( 6 ) = − 36

INVERTING A NEGATIVEEXPONENTS CHANGES ITS SIGN

€

X

− 2

=

1

X

2

€

1

X−2= X 2

DIVIDING EXPONENTS

SUBTRACT THE EXPONENTS

MULTIPLYING EXPONENTS

ADD THE EXPONENTS

€

X A • X B = X A +B

€

X A

X B= X A−B

POWER TO A POWER

€

X

2

( )

3

= X

2

( )X

2

( )X

2

( )= X

2 + 2 + 2

= X

6

THIS IS APOWERFULIDEA

Anything to the zero power

equals 1

Take A PowerTo A PowerSimplify

1. Multiply the outsidepower to the insidepowers.

2. Simplify

€

3 x3

y

7 z2

⎛

⎝

⎜

⎞

⎠

⎟

4

€

3 x3

y

2 z2

⎛

⎝

⎜

⎞

⎠

⎟

4

=3

4

x3 • 4

y1 • 4

24

z2 • 4

=3

4

x12

y4

24

z8

=81 x

12

y4

16 z8

EXPONENTS (Or POWERS are REPEATED MULTIPICATION

5 Exponent Rules1.When multiplying similar bases add exponents

2.When dividing similar bases subtract exponents

3.When inverting change exponents’ sign

4.When taking a power of a power multiply exponents

5.When taking a power of a product or quotientdistribute the outside power to the inside powers